Given a simple graph Γ, we describe a "lifting" to a 3-uniform hypergraph ϕ(Γ) that sends the complement of Γ to the complement of ϕ(Γ). We consider the effects of this lifting on cycles, complete subhypergraphs, and complete subhypergraphs missing a single hyperedge. Our results lead to natural lower bounds for some hypergraph Ramsey numbers.

For a fixed prime p Á 1 .mod 4/, we define the corresponding quartic residue graph and determine the number of triangles contained in such a graph. Our computation requires us to compute the number of pairs of consecutive quartic residues modulo p via the evaluation of certain quartic Jacobi sums.

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The most natural extensions to the law of quadratic reciprocity are the rational reciprocity laws, described using the rational residue symbol. In this article, we provide a reciprocity law from which many of the known rational reciprocity laws may be recovered by picking appropriate primitive elements for subfields of ‫(ޑ‬ζ p ). As an example, a new generalization of Burde's law is provided.

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